Combinatorics and Chandas-śāstra‑2

Ori­gins of Com­bi­na­torics in Chan­das-śās­tra

The Chan­das-śās­tra has some very inter­est­ing and intri­cate con­nec­tion with math­e­mat­ics. The word chan­das means of prosody, the sci­ence of metres. It has been esti­mat­ed by schol­ars that this Chan­das-śās­tra was com­posed by Piṅ­gala-nāga around 3rd cen­tu­ry BCE, though there could be some uncer­tain­ty in his peri­od. In his Chan­das-śās­tra, Piṅ­gala intro­duces some com­bi­na­to­r­i­al tools called pratyayas which can be employed to study the var­i­ous pos­si­ble metres in San­skrit prosody. The algo­rithms pre­sent­ed by him form the ear­li­est exam­ples of use of recur­sion in Indi­an math­e­mat­ics. In the forth­com­ing arti­cles we shall delve more into greater under­stand­ing of the var­i­ous algo­rithms or pratyayas enun­ci­at­ed by Piṅ­galācārya.

Pratyayas in Piṅgala’s Chan­das-śās­tra

In chap­ter eight of Chan­das-śās­tra, Piṅ­gala intro­duces the fol­low­ing six pratyayas:Prastāra: A pro­ce­dure by which all the pos­si­ble met­ri­cal pat­terns with a giv­en num­ber of syl­la­bles are laid out sequen­tial­ly as an array.

Saṅkhyā: The process of find­ing total num­ber of met­ri­cal pat­terns (or rows) in the prastāra.

Naṣṭa: The process of find­ing for any row, with a giv­en num­ber, the cor­re­spond­ing met­ri­cal pat­tern in the prastāra.

Uddiṣṭa: The process for find­ing, for any giv­en met­ri­cal pat­tern, the cor­re­spond­ing row num­ber in the prastāra.

Lagakriyā: The process of find­ing the num­ber of met­ri­cal forms with a giv­en num­ber of laghus (or gurus).

Adhvayo­ga: The process of find­ing the space occu­pied by the prastāra.

Algorithm#1 ~ Prastāra

Piṅ­galācārya presents the steps for con­struct­ing the prastāra, which is the first of the six pratyayas in four terse sūtras. The pro­ce­dure out­lined here helps in the ordered and con­sis­tent list­ing of all the pos­si­ble com­bi­na­tions of a n-syl­la­bled met­ri­cal pat­tern, which is tech­ni­cal­ly termed as varṇa-vṛt­ta. It is impor­tant to keep in mind that this con­struc­tion is for varṇa-vṛt­ta or met­ri­cal meters.

The fol­low­ing sūtras cor­re­spond to the pro­ce­dure of con­struct­ing prastāra.द्विकौ ग्लौ। मिश्रौ च। पृथग्लामिश्राः। वसवास्त्रिकाः ।

(छन्दःशास्त्रम् ८.२०-२३)

The steps pre­sent­ed here are essen­tial­ly the fol­low­ing:

1. Form a G, L pair. Write them one below the oth­er.

2. Insert on the right Gs [in one pair] and Ls [in anoth­er] .

3. [Repeat­ing the process] we have eight (vasavaḥ ) met­ric forms in the 3‑syl­la­ble-prastāra.

To illus­trate we shall con­sid­er the fol­low­ing pair:GL

Now we add Gs to the right of this pair and then with the same pair add Ls to the rightGGLGGLLL

The above forms the enu­mer­a­tion of a 2‑syllabled meter hence called 2‑syl­la­ble-prastāra. To form the third, we use the above set, add Gs to the right and then use the same set and add Ls to the right to obtain the 3‑syl­la­ble-prastāra. The fol­low­ing illus­tra­tion cap­tures the same.GGGLGGGLGLLGGGLLGLGLLLLL

Iter­a­tive­ly, doing the same shall lead to high­er orders. As explained in the pre­vi­ous edi­tion, sub­sti­tut­ing 0 for G and 1 for L and tak­ing the mir­ror image will lead to the mod­ern bina­ry num­bers’ ordered rep­re­sen­ta­tion.

There are also oth­er kind of prastāras for enu­mer­a­tion of mātrā-vṛt­ta or moric meters that are based on num­ber of beats/units, Tāna-Prastāra for enu­mer­a­tion of per­mu­ta­tions or tānas of svaras and Tāla-Prastāra for enu­mer­a­tion of tāla forms. All these can be broad­ly put under the body of devel­op­ment of com­bi­na­torics in India, which we shall lat­er see in this same series. Piṅ­gala has only briefly touched upon mātrā-vṛt­ta in Chap­ter IV of Chan­das-śās­tra while dis­cussing the var­i­ous forms of Āryā and Vaitālı̄ya vṛt­ta. The stud­ies of oth­er vṛt­tas hap­pened in lat­er peri­ods, which we shall learn in those respec­tive stages of this series.

Alter­na­tive Algo­rithm ~ Prastāra

In Vṛt­tarat­nākara authored by Kedāra (c.1000 CE), we find the pre­sen­ta­tion of anoth­er inge­nious algo­rithm for build­ing the n‑syl­la­ble-prastāra. पादे सर्वगुरावाद्याल्लघुं न्यस्य गुरोरधः।

यथोपरि तथा शेष भूयः कुर्यादमुं विधिम्।

ऊने दद्याद्गुरूनेव यावत्सर्वलघुर्भवेत्। ( वृत्तरत्नाकरम् ६.२‑३ )

Start with a row of Gs. Scan from the left to iden­ti­fy the first G. Place an L below that. The ele­ments to the right are brought down as they are. All the places to the left are filled up by Gs. Go on till a row of only Ls is reached.

A great advan­tage of this algo­rithm is that the ordered sequence can be built from any giv­en instance in the list­ing. The fol­low­ing illus­tra­tion presents five suc­ces­sive rows in 4‑syllable prastāra built using the algo­rithm in Vṛt­tarat­nākara.GGGLLGGLGLGLLLGLGGLL

In row 1, we have GGGL. Scan­ning from left to right, the moment we encounter a G we make it an L in the below row and retain the rest of the string, as is. Then in row 2, we have LGGL. Scan­ning from left, we iden­ti­fy the first G (sec­ond char­ac­ter), switch that to L, flip all that is there to left to Gs of that loca­tion and the char­ac­ters to the right are kept as is. Con­tin­u­ing in this fash­ion, the prastāra can be built.

It is indeed enthralling to know in depth about the rich sci­en­tif­ic her­itage of the Indi­an civ­i­liza­tion. We shall con­tin­ue to learn about the oth­er algo­rithms or pratyayas enun­ci­at­ed by Piṅ­galācārya in the fol­low­ing edi­tions.

Aum Tat Sat!